I always "got" that the Cleric was a cross between a Knight Templar and Peter Cushing's Van Helsing with spells; OK with a little bit of Archbishop Turpin from the Chanson de Roland and Bishop Odo of Bayeux thrown in too, to add to the medieval flavor. My first D&D had Crosses and Holy Water as items, and mentioned that Crosses were sovereign against Vampires regardless of their previous religious background. Honestly it never really bothered me at the time and it doesn't bother me now, even as a non-Christian in America I am still part of the cultural background and I get the context. I have played Christian characters in D&D and I don't do it as a caricature, I do it respectfully. I have also played Pagan characters the same way. When I roleplay a character I roleplay a whole person and if their faith is an important aspect of their life I respect that.
My last post went off a little half cocked because I had just read 3 or 4 different blogs talking about Christianity and it's compatibility with D&D and why that is either a great idea or a crap idea. I was up late when I wrote it and just wanted to say what I had to say on the subject; I thought I could be quick and to the point. I think it's obvious, although never made explicit, that D&D was designed with a medieval European Christian background in mind and that ultimately TSR removed references to Christianity to protect Christians. The ones that felt uneasy about having their religion portrayed in an explicitly fantasy game. The so called "Satanic Panic" and BADD must have caught them completely unexpectedly since they were trying to do good Christians a favor.
In my experience, the "Satanic Panic" of the 1980s was based on a mostly Catholic/Protestant divide, I say mostly because it was only the more fundamentalist denominations that really bought into the whole thing. Catholics were mostly pretty cool with D&D. More mainstream Protestant denominations didn't seem to have a problem with it. I live in the North-East US though, so your mileage may vary. I don't know how Orthodox Christians felt about it then or now. Jews seem to be OK with D&D. I don't know enough Muslims to get an accurate sample. No other religion would be concerned with Satan in the least.
What muddied the waters was the publication of Gods, Demigods and Heroes; superseded in AD&D by Deities & Demigods. I do not own, nor have I read Gods, Demigods and Heroes, but I do own Deities & Demigods (sadly lacking in Cthulu and Elric) and I assume they are pretty similar, designed to give the average American a "fictional" pantheon to play with so as to protect the feelings of Christians that might be uncomfortable with their religion being implied in a fantasy context. EGG goes one better in the pages of The Dragon by offering up entire pantheons of his own devising of entirely fictional deities for Greyhawk, which I am way more cool with.
I understand that in the 1970s and 1980s we didn't have the cultural sensitivity and awareness that we have now, but damn, between Buddhism, Hinduism and Shinto you are coving something like 1/2 of the world's population. Throw in traditional practitioners of Native American religions, Sami Shamanists (a religion closely related to the "Finnish Mythos"), Wiccans, and religious reconstructionists of various types and pretty much the only unworshipped gods in my copy of Deities & Demigods are the "Newhon Mythos" and "Nonhuman Deities". The fact that the powers that be at TSR at the time, presumably EGG and I have no idea who else, sat down and talked it over and felt uneasy enough that they had to remove explicit references to Christianity from the game but give fictional alternatives speaks volumes. The hubris expressed by them in only eliminating references to Abrahamic religions also speaks volumes about our culture at the time.
I can understand seriously minority religions being fodder for the Deities & Demigods book, but Buddhism, Hinduism and Shinto have between them Billions of adherents, just not in the US (although Hawaii has a substantial Buddhist minority, Hawaii usually falls outside the spectrum of consideration for most decisions made concerning the US as a whole); I really think that maybe they could have, even in the caveman 1970s and 80s, considered the implications there.
Now, Deities & Demigods pretty much told you that you HAD to be a Pagan in AD&D, it's one of those things where if they hadn't mentioned it, probably nobody would have thought of it, but since they did it got set in stone. They gave you a nice bunch of pantheons to choose from; although they treated polytheism the way you would expect a bunch of gamers from a monotheistic background to treat it; choose a pantheon, then pick one deity from that pantheon as a patron and worship him or her pretty much monotheisticly. That's totally NOT how polytheism works, just an FYI.
In the wild and woolly days after the release of the Deities & Demigods book pretty much everything in it was fair game, at least in my neck of the woods. So every player would flip through the book and pick a patron god (and check out the goddesses boobs) from whatever pantheon struck his fancy (yes, we were mostly boys then, with a few notable exceptions) and away we'd go.
That said, to be honest, in my longest running AD&D campaign in the 1980s I was a player, and the DM didn't own the Deities & Demigods book, neither did I until about 1985-ish, and religion was never an issue at all in his game. We had a town that was our home base, there was a tavern and a church and some other shops. Implied Christianity. Hell, even the Dwarven Cleric miniature from Grenadier has a Cross, and we used those official gold box Grenadier AD&D Miniatures almost exclusively. What didn't come in boxed sets from Grenadier, the only miniatures you could find in central New York back then, came from toy sets- dinosaurs, animals and stuff like that. I even found in a bargain bin at a local department store a plastic Bulette and a plastic Rust Monster, IDed from their pictures in the Monster Manual. I really wish I knew what happened to them.
Later on responsible DMs like myself set limits on what pantheons you could choose gods from, the favorite in my experience was the Norse pantheon. I assume this is because of a combination of ass-kicking gods, easily pronounced names (for the most part), Thor comics, and cultural familiarity. I have run games using Norse gods, Celtic gods, Roman Gods, Pantheons of my own devising based on pure fantasy, Pantheons of my own devising based on Indo-European commonalities, the Celestial Bureaucracy, Path of Enlightenment and Way of 8 Million gods in Kara-Tur and Greyhawk Gods in the World of Greyhawk. My own Garnia campaign uses a mixture of real world religions and fantasy religions and extrapolated religions based on real world religions that became extinct. I expect that my players are going to be mature enough to be respectful of religion in the context of the game and that they will be respectful of religion outside of the game too.
We have a diversity of religions at my game table that would probably have never happened back in the 1980s. Back in the 1980s in America you had essentially three or four choices of religion; Catholic, Protestant, Agnostic, or Atheist. Yes, I count no religion as a religion here, particularly when some Atheists get really preachy about it. At my game we have at least two different sects of Christianity, an Atheist, an Agnostic or two, and a couple of Heathen Asatruar; and those are just the ones I know the religious preferences of. My house, at game time, is a religious neutral zone. The characters currently are all adherents of various sects of fantasy versions of Shinto or Buddhism because we're playing an Oriental Adventures campaign set in Kozakura. In my last campaign they were all adherents of an extrapolated Celtic pantheon that included heroes that had been deified over the course of my 30+ years of DMing in Garnia. That pantheon also includes Christ.